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What the Nash Equilibrium Can and Cannot Tell Us

The mathematician John Forbes Nash of 'A Beautiful Mind' died in a car crash. What is his legacy?
May 25, 2015, 5:00pm
Nash. Image: Peter Badge/Wikipedia

Yesterday John Nash died in a car crash in New Jersey. Although there is no doubting his genius and the importance of his work, he himself advised caution when applying his theories to real world situations.

At the turn of the 1950s the greatest minds of America gathered in a secret location (which may or may not have looked like a scene out of Dr. Strangelove). We now know this secret location was the RAND. Corporation's HQ in sunny Santa Monica, and in attendance was John Forbes Nash.

Nash had been selected to join the global policy think tank in order to advise the government on how to overcome the menace of communism posed by the Soviet Union. His inclusion was based on the Ph.D thesis he had completed just months before. His dissertation was titled "Non-Cooperative Games" and represented a major development in the understanding of game theory.

Only 21 by the time his paper was published, Nash had conceived an idea that would capture the attention of the western intellectual elite and would filter down throughout society, bringing about a collective new way of thinking. In 1994, 44 years later, he would win the Nobel Prize for this idea (and eight years after that he became the subject of award winning movie, A Beautiful Mind).

"Nash equilibrium" describes a situation where each player in a game is making the best decision for themselves, whilst taking into account the decisions being made by each other player. In other words, there is no incentive for any individual player to change their strategy.

Once you start looking, Nash equilibrium emerges everywhere

An obvious example is nuclear arms. Two warring countries with nuclear weapons know that it is in their best interest not to use their weapons. If they were to press the big red button the other country would retaliate and the result would be mutually assured destruction. A crude simplification but nevertheless we have a Nash equilibrium—truce.

Once you start looking, Nash equilibrium emerges everywhere. Business and economics often brings about Nash equilibrium, especially when the market is dominated by just a few traders. The Phoebus cartel was an agreement between a cohort of companies involved in the manufacturing and selling of light bulbs. In one of the earliest examples of planned obsolescence, the cartel eradicated outside competition and enforced rules that the duration of any one light bulb must not exceed 1,000 hours, thus keeping demand high and costs low.

In an open market with a large number of traders, companies may decide it is in their best interest to try and outcompete other manufacturers by producing longer lasting bulbs. When competition becomes more fierce, manufacturers may lose sight of the team ethic and instead pursue their own independent success—creating a saturated market of bulbs that never burn out. This is not profitable for any individual company, nor the market in general—but is of course beneficial to the consumer.

It becomes patently clear why the Nash equilibrium was such a crucial milestone, not just in economics but society as a whole. Nash's mathematical ideas would underpin the ongoing Cold War narrative even after he was removed from RAND due to being arrested on the suspicion of engaging in homosexual behaviour. At the same time he was also descending into the delusional world of schizophrenia, which lead to him spending the next few decades being involuntarily incarcerated in mental institutes.

The application of game theory assumes that people are rational maximizers, and so there is a sad irony in the fact Nash was suffering from insanity when he derived his theories. However, this irony may be illustrative of the limitations of his work; emotion and mental instability are factors which simply don't come into these equations.

Nash considered himself a pure mathematician

There is always a tendency to apply pure mathematical and scientific theory to our own lives but it doesn't always fit. We are all complicated and it is hard to ever imagine somebody writing an equation that describes our innermost thoughts and desires.

It could be argued from an economic perspective that businesses behave in a relatively unemotional capacity. This may well be the case, and I am sure there are plenty of useful applications of Nash's work. However, it is worth noting that Nash considered himself a pure mathematician, skeptical of how far his ideas could be applied to humanity. Throughout the time he was being grafted into Cold War think tanks, he always expressed his interest in equations not politics.

It is a testament to Nash's resilience that such a troubled and unwell man lived to the age of 86. In his death, we should be reminded that we are all human. Our behaviour is complex and our desires are driven by a multitude of factors from compassion to greed, rational to irrational. Our psychology cannot be explained by general mathematical equations—but this was never what Nash intended.

In her biography of Nash, Sylvia Nasar retells a speech made at Princeton during a post-Nobel celebratory party where Nash said. "Game theory is a subject of great intrinsic intellectual interest," he reportedly said, "that the world wishes to imagine can be of some utility."